Sharp nonlogarithmic fundamental-solution decay

Prove or disprove that every C^1 weak solution of the p-Laplace Lane–Emden equation satisfying the small self-similar decay hypothesis in Proposition 4.1 obeys the sharp bound $u(x)=O(|x|^{-gamma})$ as $|x|\to\infty$, without a logarithmic factor.

Background

The comparison bootstrap improves small self-similar decay to a p-harmonic decay rate multiplied by an arbitrarily small positive power of log|x|. The logarithmic factor arises from the iteration and is harmless for the finite-energy argument, but the paper does not establish the genuinely sharp bound at the fundamental-solution rate. Determining whether the logarithm can be removed is left unresolved.

References

Since $\theta_\tau\to0$ as $\tau\to\infty$, the exponent $\theta$ in eq_pharmonic may be replaced by any positive number, at the cost of a larger constant. It cannot be replaced by $0$ this way, because $C_\tau$ grows with $\tau$, and we do not know whether $u(x)=O(|x|{-\gamma})$ holds.

eq_pharmonic:

∣u(x)∣≤C ∣x∣−γ(log⁡∣x∣)θfor ∣x∣≥R∗.|u(x)|\le C\,|x|^{-\gamma}\left(\log|x|\right)^{\theta}\qquad\text{for } |x|\ge R_* .

— Sharp Liouville thresholds and endpoint rigidity for finite Morse index solutions of the $p$-Laplace Lane--Emden equation  (2610.02751 - Le, 2 Oct 2026) in Remark following Corollary 5.2 in Section 5